Introduction to the Total Differential

1. Formula
   Observation

   The instructor writes $z = f(x,y)$ on the whiteboard.

2. Audio
   Observation

   He says, 'let's say that we have z and it's a function of two variables f of x y'.

z

Dependent variable or output of the function f.

Real numbers

1. Formula
   Observation

   The instructor writes $z = f(x,y)$ on the whiteboard.

2. Audio
   Observation

   He refers to it as 'a function of two variables f of x y'.

f

A function of two variables.

Function from $R^2$ to R

1. Formula
   Observation

   The instructor writes $z = f(x,y)$ on the whiteboard.

2. Audio
   Observation

   He refers to x as one of the two variables.

x

First independent variable of the function f.

Real numbers

1. Formula
   Observation

   The instructor writes $z = f(x,y)$ on the whiteboard.

2. Audio
   Observation

   He refers to y as one of the two variables.

y

Second independent variable of the function f.

Real numbers

1. Formula
   Observation

   The instructor writes dz = ... on the whiteboard.

2. Audio
   Observation

   He says, 'It's dz and it's equal to...'

dz

The total differential of z.

Real numbers

1. Formula
   Observation

   The instructor writes $\frac{\partial f}{\partial x}$ on the whiteboard.

2. Audio
   Observation

   He says, 'del f del x'.

$\frac{\partial f}{\partial x}$

The partial derivative of f with respect to x.

Real numbers

1. Formula
   Observation

   The instructor writes dx on the whiteboard.

2. Audio
   Observation

   He says, 'dx'.

dx

Differential of the independent variable x.

Real numbers

1. Formula
   Observation

   The instructor writes $\frac{\partial f}{\partial y}$ on the whiteboard.

2. Audio
   Observation

   He says, 'plus del f del y'.

$\frac{\partial f}{\partial y}$

The partial derivative of f with respect to y.

Real numbers

1. Formula
   Observation

   The instructor writes dy on the whiteboard.

2. Audio
   Observation

   He says, 'dy'.

dy

Differential of the independent variable y.

Real numbers

1. Audio
   Observation

   He says, 'dz actually approximates delta z which is the change in z'.

1. The symbol $\Delta z$ is not written on the board, only spoken.

$\Delta z$

The actual change in the value of z.

Real numbers

1. Formula
   Observation

   The instructor writes 'The total differential of $z = f(x,y)$ is $dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$'.

2. Audio
   Observation

   He explains the formula term by term.

For a function $z = f(x,y)$ of two variables, the total differential, denoted dz, is defined as the sum of the partial derivative of f with respect to x multiplied by dx, and the partial derivative of f with respect to y multiplied by dy.

1. z is a function of two variables, x and y.

2. The partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist.

1. Partial Derivative Concept

1. Audio
   Observation

   He says, 'when you take the partial with respect to x, all of your y's are constants'.

When taking the partial derivative of a multivariable function with respect to one variable, the other variables are treated as constants.

1. The function has more than one variable.

1. Audio
   Observation

   He mentions exact differential equations and approximations.

The total differential is used in solving exact differential equations and for approximating the change in a function's value ($\Delta z \approx dz$).

1. Definition of the Total Differential

1. Audio
   Observation

   He says, 'Some people call it the total derivative as well, but I believe this is the most common name'.

The concept is commonly called the 'total differential', though some may refer to it as the 'total derivative'.

No quantifiers are stated in the source.

1. Formula
   Observation

   The instructor writes the steps for finding the total differential of $z = x^2y^2$.

1. Expression

   $$z = x^2y^2$$
   Explanation

   Given function.

   Justification

   Problem statement.

   Shown in the video
2. Expression

   $$\frac{\partial z}{\partial x} = 2xy^2$$
   Explanation

   Take the partial derivative with respect to x, treating y as a constant.

   Justification

   Power rule and constant multiple rule for partial derivatives.

   Shown in the video
3. Expression

   $$\frac{\partial z}{\partial y} = 2x^2y$$
   Explanation

   Take the partial derivative with respect to y, treating x as a constant.

   Justification

   Power rule and constant multiple rule for partial derivatives.

   Shown in the video
4. Expression

   $$dz = 2xy^2 dx + 2x^2y dy$$
   Explanation

   Substitute the partial derivatives into the total differential formula.

   Justification

   Definition of the total differential.

   Shown in the video

The total differential of $z = x^2y^2$ is $dz = 2$xy^$2 dx + 2x^2y dy$.

1. Formula
   Observation

   The instructor writes 'Ex $z = x^2y^2$' and solves for dz.

Find the total differential of the function $z = x^2y^2$.

1. $z = x^2y^2$

Compute dz.

1. Expression

   $$\frac{\partial z}{\partial x} = 2xy^2$$
   Explanation

   Differentiate with respect to x, holding y constant.

   Justification

   Rules of partial differentiation.

   Shown in the video
2. Expression

   $$\frac{\partial z}{\partial y} = 2x^2y$$
   Explanation

   Differentiate with respect to y, holding x constant.

   Justification

   Rules of partial differentiation.

   Shown in the video
3. Expression

   $$dz = 2xy^2 dx + 2x^2y dy$$
   Explanation

   Apply the formula for the total differential.

   Justification

   Definition of the total differential.

   Shown in the video

$dz = 2$xy^$2 dx + 2x^2y dy$

The result matches the standard formula for the total differential applied to the given function.

1. Animation
   Observation

   The instructor writes the definition of the total differential on the whiteboard step-by-step.

1. Whiteboard

2. Marker

3. Instructor

1. Text appears on the whiteboard: 'The total differential of $z = f(x,y)$ is $dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$'.

1. The instructor remains in front of the whiteboard.

Visually presents the mathematical definition being discussed.

1. Animation
   Observation

   The instructor writes an example problem and its solution on the whiteboard.

1. Whiteboard

2. Marker

3. Instructor

1. Text appears on the whiteboard: 'Ex $z = x^2y^2$' followed by '$dz = 2$xy^$2 dx + 2x^2y dy$'.

1. The instructor remains in front of the whiteboard.

Demonstrates the application of the total differential formula to a specific function.

1. Audio
   Observation

   He clarifies the naming convention.

Students might be confused if they hear the term 'total derivative'.

The instructor notes that while 'total differential' is the most common name, some people also call it the 'total derivative'.

1. Formula
   Observation

   The formula for dz uses $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$.

Understanding partial derivatives is necessary to compute the total differential.

1. Audio
   Observation

   He discusses applications like exact differential equations and approximations.

The concept of the total differential is applied in solving certain types of differential equations and in making approximations.

1. Formula
   Observation

   The definition is written on the board.

1. Definition of the Total Differential

1. Formula
   Observation

   An example is solved on the board.

1. Example: Finding the Total Differential

1. Audio
   Observation

   Applications are discussed verbally.

1. Applications of the Total Differential